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In this paper, we introduce Dependency Dialogue Acts (DDA), a novel framework for capturing the structure of speaker-intentions in multi-party dialogues. DDA combines and adapts features from existing dialogue annotation frameworks, and emphasizes the multi-relational response structure of dialogues in addition to the dialogue acts and rhetorical relations. It represents the functional, discourse, and response structure in multi-party multi-threaded conversations. A few key features distinguish DDA from existing dialogue annotation frameworks such as SWBD-DAMSL and the ISO 24617-2 standard. First, DDA prioritizes the relational structure of the dialogue units and the dialog context, annotating both dialog acts and rhetorical relations as response relations to particular utterances. Second, DDA embraces overloading in dialogues, encouraging annotators to specify multiple response relations and dialog acts for each dialog unit. Lastly, DDA places an emphasis on adequately capturing how a speaker is using the full dialog context to plan and organize their speech. With these features, DDA is highly expressive and recall-oriented with regard to conversation dynamics between multiple speakers. In what follows, we present the DDA annotation framework and case studies annotating DDA structures in multi-party, multi-threaded conversations.